Why did Lot's Tefilah save Tzo'ar, while Avraham's Tefilah did not save it?
Moshav Zekenim: Lot asked only for Tzo'ar. All of Avraham's requests were to save all five cities. 1
Ramban (to 19:12): Perhaps this was in honor of his hospitality. It is proper for Sheluchim to save the entire house of their hosts, like we find that Yehoshua's Sheluchim saved Rachav's family.
Refer to 18:28:1:2; refer to 18:29:0.1:2.
QUESTIONS ON RASHI
Rashi writes: "'Hofki' - Hofech Ani (I am [not] overturning])." What is Rashi explaining?
Gur Aryeh: The final Yud, indicating the first person, relates to the subject of the verb, not its object. (I.e. the angel who was speaking would be doing the overturning, not being overturned).